Structure Theorems for the Symmetric Groups Acting on its Natural Module

TL;DR

This paper provides an explicit structure theorem for the symmetric group acting on the symmetric algebra of its natural module, detailing the module decomposition in terms of elementary symmetric polynomials and monomial representations.

Contribution

It introduces a new explicit isomorphism for the symmetric group's action on the symmetric algebra, extending previous understanding of module structures.

Findings

Decomposition of the symmetric algebra into modules involving elementary symmetric polynomials.

Explicit isomorphism of $kG$-modules for the symmetric group action.

Connection to monomial representations as modules $V_I$.

Abstract

This paper gives an explicit structure theorem for the symmetric group acting on the symmetric algebra of its natural module. Let $G$ be the symmetric group on $x_1,..., x_n$ and let $d_i$ be the $i^{\text{th}}$ elementary symmetric polynomial in the $x_i$'s. We show that if we take monomial representations discussed in \cite[Section 3]{Kemper} to be the modules $V_I$, then we have an isomorphism of $kG$-modules $k[x_1,..., x_n] \cong \Oplus_{\{n\} \subseteq I \subseteq [n]} k[d_I] \otimes_k V_I$.